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Author Hstad, Johan ♦ Nslund, Mats
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2004
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Cryptography ♦ RSA-encryption ♦ Bit-security ♦ Complexity ♦ Discrete logarithms
Abstract We study the security of individual bits in an RSA encrypted message $E_{N}(x).$ We show that given $E_{N}(x),$ predicting any single bit in $\textit{x}$ with only a nonnegligible advantage over the trivial guessing strategy, is (through a polynomial-time reduction) as hard as breaking RSA. Moreover, we prove that blocks of $\textit{O}(log$ log $\textit{N})$ bits of $\textit{x}$ are computationally indistinguishable from random bits. The results carry over to the Rabin encryption scheme.Considering the discrete exponentiation function $g^{x}$ modulo $\textit{p},$ with probability 1 ™ $\textit{o}(1)$ over random choices of the prime $\textit{p},$ the analog results are demonstrated. The results do not rely on group representation, and therefore applies to general cyclic groups as well. Finally, we prove that the bits of $\textit{ax}$ + $\textit{b}$ modulo $\textit{p}$ give hard core predicates for any one-way function $\textit{f}.All$ our results follow from a general result on the chosen multiplier hidden number problem: given an integer $\textit{N},$ and access to an algorithm $P_{x}$ that on input a random $\textit{a}$ ∈ $Z_{N},$ returns a guess of the $\textit{i}th$ bit of $\textit{ax}$ mod $\textit{N},$ recover $\textit{x}.$ We show that for any $\textit{i},$ if $P_{x}$ has at least a nonnegligible advantage in predicting the $\textit{i}th$ bit, we either recover $\textit{x},$ or, obtain a nontrivial factor of $\textit{N}$ in polynomial time. The result also extends to prove the results about simultaneous security of blocks of $\textit{O}(log$ log $\textit{N})$ bits.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2004-03-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 51
Issue Number 2
Page Count 44
Starting Page 187
Ending Page 230


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Source: ACM Digital Library